Unit 1: Numbers & Numerical Applications

Modulo Arithmetic, Mixtures, Boats & Streams, Partnerships

1. Modulo Arithmetic

Definition If $$a$$is divided by a positive integer$$n$$, it leaves a remainder $$r$$. This is written as:
$$a \pmod n = r$$where$$0 \le r < n$$.
Congruence Modulo Two integers $$a$$and$$b$$are congruent modulo$$n$$if they leave the same remainder when divided by$$n$$.
$$a \equiv b \pmod n \iff n \text{ divides } (a – b)$$
Properties of Addition $$(a + b) \pmod n = \big((a \pmod n) + (b \pmod n)\big) \pmod n$$
Properties of Multiplication $$(a \times b) \pmod n = \big((a \pmod n) \times (b \pmod n)\big) \pmod n$$
Power Property If $$a \equiv b \pmod n$$, then $$a^k \equiv b^k \pmod n$$for any positive integer$$k$$.

2. Allegation and Mixture

Concept: Used to find the ratio in which two ingredients of different prices (or attributes) must be mixed to produce a mixture of a desired mean price.

Variables $$C$$ = Cost of Cheaper ingredient
$$D$$ = Cost of Dearer (Expensive) ingredient
$$M$$ = Mean price of the mixture
Rule of Allegation $$\frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \frac{D – M}{M – C}$$
Repeated Dilution If $$x$$is the initial quantity of pure liquid and$$y$$units are taken out and replaced by water$$n$$ times:
$$\frac{\text{Liquid Left}}{\text{Total Capacity}} = \left( 1 – \frac{y}{x} \right)^n$$

3. Boats and Streams

Variables $$u$$ = Speed of boat in still water
$$v$$ = Speed of stream (current)
Downstream Speed ($$D_s$$) $$D_s = u + v$$ (Along the flow)
Upstream Speed ($$U_s$$) $$U_s = u – v$$ (Against the flow)
Finding $$u$$and$$v$$ Boat Speed: $$u = \frac{D_s + U_s}{2}$$
Stream Speed: $$v = \frac{D_s – U_s}{2}$$

4. Pipes and Cisterns

Basic Principle If a pipe fills a tank in $$x$$hours, part filled in 1 hour =$$1/x$$.
If a pipe empties a tank in $$y$$hours, part emptied in 1 hour =$$1/y$$.
Net Work (Inlet + Outlet) If both pipes work together:
Net part filled in 1 hour = $$\frac{1}{x} – \frac{1}{y}$$(assuming$$x < y$$)

5. Races and Games

“A gives B a start of $$x$$ meters” A runs the full distance $$D$$, while B runs $$D – x$$.
“A beats B by $$t$$ seconds” A finishes the race $$t$$seconds before B. (Time Difference =$$t$$)
Dead Heat Both participants reach the finishing point at the exact same time.

6. Numerical Inequalities

Multiplication/Division Rule If $$a > b$$and$$k < 0$$ (negative), then:
$$a \times k < b \times k$$ (Inequality sign reverses)
Reciprocal Rule If $$a > b > 0$$, then:
$$\frac{1}{a} < \frac{1}{b}$$
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